Communications in Mathematical Sciences

Volume 8 (2010)

Number 3

Special Issue: Mathematical Issues of Complex Fluids

Local existence and uniqueness of the dynamical equations of an incompressible membrane in two-dimensional space

Pages: 783 – 796



Dan Hu

Peng Song

Pingwen Zhang


The dynamics of a membrane is a coupled system of a moving elastic surface and an incompressible membrane fluid. The difficulties in analyzing such a system include the nonlinearity of the curved space (geometric nonlinearity), the nonlinearity of the fluid dynamics (fluid nonlinearity), and the coupling to the surface incompressibility. In the two-dimensional case, the fluid vanishes and the system reduces to a coupling of a wave equation and an elliptic equation. Here we prove the local existence and uniqueness of the solution to the system by constructing a suitable discrete scheme and proving the compactness of the discrete solutions. The risk of blowing up due to the geometric nonlinearity is overcome by the bending elasticity.


Membrane; incompressible; existence; uniqueness; bending elasticity

2010 Mathematics Subject Classification

35M13, 65M12, 92C17

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